Choosing the right neural network settings
Easy ones first:


Activations are going to be RELU all the way down, until your final softmax layer (win probability?) (because empirically, RELU does great on most problems, except when your model is an RNN, and it makes the gradient explode, or in the final layer of a regression model - numerical stability etc.).
You probably to structure this with some layers of convolutions with max pooling between them, then 1 or 2 fully connected layers (FC) near the end (because if you only have FC layers, you probably won't have enough data to train them)
Well worth trying some 1D convolutions (which cleverly combine channels from convolutions created by previous layers).
Learning rate: take the SGD default to begin with, then tune later. Problem dependent, I'm afraid!  The returns to tuning can be large though (my kids twigged to this quite quickly when playing with a toy problem).


Now the hard bit - encoding your input:


Categorical encoding using a single integer per board position could cause your model some grief (it is an input that "looks like" a real number, but of course it isn't, and values that seem numerically close may represent pieces with radically different abilities (perhaps the code for King is 1 and Queen is 2 and Bishop is 3, but all those pieces have such different attributes).
I would strongly consider thinking of each piece/player combo as a "colour channel" - a 64 cell grid where each value is either zero or one (so, one channel for White's pawns, another for White's knights, and so on).


Finally, those labels of yours: do think about what you'll do with drawn games - perhaps you have 3 possible outcomes, not 2?

We would all be fascinated to hear how you get on - I hope you'll write your work up in some form (and that you'll come back and complain/praise our advice, as appropriate!).